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Cubicity, Degeneracy, and Crossing Number | | Abhijin Adiga
; L. Sunil Chandran
; Rogers Mathew
; | | Date: |
26 May 2011 | | Abstract: | A $k$-box $B=(R_1,R_2,...,R_k)$, where each $R_i$ is a closed interval on the
real line, is defined to be the Cartesian product $R_1 imes R_2 imes... imes
R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube.
emph{Boxicity} of a graph $G$, denoted as $oxi(G)$, is the minimum integer
$k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the
emph{cubicity} of $G$, denoted as $cubi(G)$, is the minimum integer $k$ such
that $G$ is an intersection graph of $k$-cubes.
It was shown by Chandran et al. that, for a graph $G$ with maximum degree
$Delta$, $cubi(G)leq lceil 4(Delta +1)ln n
ceil$. In this paper, we show
that, for a $k$-degenerate graph $G$, $cubi(G) leq (k+2) lceil 2e log n
ceil$. Since $k$ is at most $Delta$ and can be much lower, this clearly is a
stronger result. We also give an efficient deterministic algorithm that runs in
$O(n^2k)$ time to output a $8k(lceil 2.42 log n
ceil + 1)$ dimensional cube
representation for $G$.
An important consequence of the above result is that if the crossing number
of a graph $G$ is $t$, then $oxi(G)$ is $O(t^{1/4}{lceillog
t
ceil}^{3/4})$ . This bound is tight upto a factor of $O((log t)^{3/4})$.
Let $(mathcal{P},leq)$ be a partially ordered set and let $G_{mathcal{P}}$
denote its underlying comparability graph. Let $dim(mathcal{P})$ denote the
emph{poset dimension} of $mathcal{P}$. Another interesting consequence of our
result is to show that $dim(mathcal{P}) leq 2(k+2) lceil 2e log n
ceil$,
where $k$ denotes the degeneracy of $G_{mathcal{P}}$. Also, we get a
deterministic algorithm that runs in $O(n^2k)$ time to construct a $16k(lceil
2.42 log n
ceil + 1)$ sized realizer for $mathcal{P}$. | | Source: | arXiv, 1105.5225 | | Services: | Forum | Review | PDF | Favorites | |
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